As shown in Section REF , our results suggest that the
processes linked to AGNs may change the properties of their host galaxies.
If we want to investigate whether or not galaxy environment
has played an important role in these processes, galaxies in the
control sample should be statistically similar to the progenitors
of the AGN hosts (i.e. the host galaxies before the onset of the AGN)
rather than the host galaxies on the AGN duty cycle.
This suggests that we should select, as our comparison
sample, normal galaxies that have the same properties
as the AGN host galaxies before the onset of AGN.
This is not straightforward to do but our results provide some hints.
Since AGN hosts and star-forming galaxies share the same SHMR,
controlling stellar mass is equivalent to controlling halo mass.
Since the average formation history of central galaxies
is determined by the host halo mass, star-forming galaxies controlled
in stellar mass thus provide a comparison sample that we need to
investigate whether or not the host galaxies of AGNs are special
in their environment and evolutionary stages relative to the average
population of galaxies. According to \(N\) -body simulations, halos of
\(10^{12}\, h^{-1}{\rm M_\odot }\)  at \(z=0\)  assembled half of their mass at \(z\sim 1\) 
[1]}, and so the growth timescale for a
\(10^{12}\, h^{-1}{\rm M_\odot }\)  halo is typically about 7 Gyrs.
This time scale is much longer than the time scale
for the evolution between star-forming galaxies and AGNs,
which is about 1-2 Gyrs according to the values of
\(D_n4000\)  of AGN hosts and star-forming galaxies. This
indicates that controlling halo mass also provides a
stable reference to investigate evolution in star formation and
AGN activities.
